It's not a switch, it's a trigger wire that is spliced into the backup light circuit. When the backup lights come on it puts +12volts onto that trigger wire which activates the camera. Sounds like a loose connection or a faulty camera system. Don't know where the wiring connections are made.
Do you have a make and model for the system? Do you have a manual for it? If so there probably is a wiring diagram for the system so you can find which wire is the trigger.
Also, don't rule out just a loose ground wire. Loose ground wires account for a majority of electrical problems and are an easy fix.